How Does Systematic Risk Impact US Credit Spreads? A Copula Study

It is well known that some relationship between systematic risk and credit risk prevails in financial markets. In our study, S&P 500 stock index return is our market risk proxy whereas credit spreads represent our credit risk proxy as a function of maturity, rating and economic sector. We address the problem of studying the joint distributions and evolutions of S&P 500 return and credit spreads. Graphical and non parametric statistical analysis (i.e.: Kendall’s tau and Spearman’s rho) show that such bivariate distributions are asymmetric with some negative relationship between S&P 500 return and credit spreads. In-deed, credit spreads widen when S&P 500 return decreases or drops under some given level. We investigate then this stylized fact using copula functions to characterize observed dependence structures between S&P 500 return and credit spreads. We focus at least on one parameter copulas and at most on one parameter Archimedean copulas, namely Gumbel, FGM, Frank and Clayton copula functions. Starting from empirical Kendall’s tau observed for each bivariate dependence structure, we induce parameter values for each copula type function belonging to our copulas set. Finally, we exhibit optimal characterizations for such dependence structures and use the optimal selected copulas to achieve a scenario analysis among which stress testing.systematic risk credit risk copulas Archimedean copulas stress testing

Price Calibration of basket default swap: Evidence from Japanese market

The aim of this paper is the price calibration of basket default swap from Japanese market data. The value of this instruments depend on the number of factors including credit rating of the obligors in the basket, recovery rates, intensity of default, basket size and the correlation of obligors in the basket. A fundamental part of the pricing framework is the estimation of the instantaneous default probabilities for each obligor. Because default probabilities depend on the credit quality of the considered obligor, well-calibrated credit curves are a main ingredient for constructing default times. The calibration of credit curves take into account internal information on credit migrations and default history. We refer to Japan Credit Rating Agency to obtain rating transition matrix and cumulative default rates. Default risk is often considered as a rare-event and then, many studies have shown that many distributions have fatter tails than those captured by the normal distribution. Subsequently, the choice of copula and the choice of procedures for rare-event simulation govern the pricing of basket credit derivatives. Joshi and Kainth (2004) introduced an Importance Sampling technique for rare-event that forces a predetermined number of defaults to occur on each path. We consider using Gaussian copula and t-student copula and study their impact on basket credit derivative prices. We will present an application of the Canonical Maximum Likelihood Method (CML) for calibrating t-student copula to Japanese market data.Basket Default Swaps, Credit Curve, Monte Carlo method, Gaussian copula, t-student copula, Japanese market data, CML, Importance Sampling

Copula models for epidemiological research and practice

Investigating associations between random variables (rvs) is one of many topics in the heart of statistical science. Graphical displays show emerging patterns between rvs, and the strength of their association is conventionally quantified via correlation coefficients. When two or more of these rvs are thought of as outcomes, their association is governed by a joint probability distribution function (pdf). When the joint pdf is bivariate normal, scalar correlation coefficients will produce a satisfactory summary of the association, otherwise alternative measures are needed. Local dependence functions, together with their corresponding graphical displays, quantify and show how the strength of the association varies across the span of the data. Additionally, the multivariate distribution function can be explicitly formulated and explored. Copulas model joint distributions of varying shapes by combining the separate (univariate) marginal cumulative distribution functions of each rv under a specified correlation structure. Copula models can be used to analyse complex relationships and incorporate covariates into their parameters. Therefore, they offer increased flexibility in modelling dependence between rvs. Copula models may also be used to construct bivariate analogues of centiles, an application for which few references are available in the literature though it is of particular interest for many paediatric applications. Population centiles are widely used to highlight children or adults who have unusual univariate outcomes. Whilst the methodology for the construction of univariate centiles is well established there has been very little work in the area of bivariate analogues of centiles where two outcomes are jointly considered. Conditional models can increase the efficiency of centile analogues in detection of individuals who require some form of intervention. Such adjustments can be readily incorporated into the modelling of the marginal distributions and of the dependence parameter within the copula model

Modeling Tropical Cyclone Storm Surge and Wind Induced Risk along the Bay of Bengal Coastline Using a Statistical Copula

High winds, torrential rain, and storm surges from tropical cyclones cause massive destruction to property and cost the lives of many people. Among the coastal areas affected by these major natural calamities, the coastline of the Bay of Bengal (BoB) ranks as one of the most susceptible to tropical cyclone storm surge risk due to its geographical setting and population density, Bangladesh suffers the most. The purpose of this study is to describe the relationship between storm surge at the BoB and peak reported wind and describe the dependency structure between wind speeds and storm surges at that location. Various models have been developed to predict storm surge in this region but none of them quantify statistical risk with empirical data. This research demonstrates a methodology for estimating the return period of the joint hazard based on a bivariate copula model. An Archimedean Gumbel copula with Weibull and normal margins is specified for the result the coast of BoB can expect a cyclone with peak reported winds of at least 24 m s−1 and surge heights of at least 4.0 m, on average, once every 3.2 years (2.7–3.8). The BoB can expect peak reported winds of 62 m s−1 and surge heights of at least 8.0 m, on average, once every 115.4 years (55.8–381.1). In this ocean basin, surge heights are comparably higher when compared to other ocean basins. Application of the copula will mitigate future threats of storm surge impacts on coastal communities of the BoB

Essays on Quantitative Risk Management

The costly lessons from global crisis in the past decade reinforce the importance as well as challenges of risk management. This thesis explores several core concepts of quantitative risk management and provides further insight. We start with rating migration risk and propose a Mixture of Markov Chains (MMC) model to account for stochastic business cycle effects in credit rating migration risk. The model shows superior in-sample estimation and out-of-sample predication than its rivals. Compared with the naive approach the economic application suggests banks with MMC estimator will increase capital requirement in economic expansion and free up capital during recession hence it is aligned with Basel III macroprudential imitative by reducing the recession-vs-expansion gap in capital buffers. Subsequently we move to the key concept of dependence by investigating the importance of dynamic linkages between credit and equity markets. We propose a flexible regime-switching copula model to explore the dynamics of dependence and possible structure breaks with special consideration on tail dependence. The study reveals a high-dependence regime that coincides with the recent financial crisis. The backtesting results acknowledge the new model's superiority on out-of-sample VaR forecasting over purely dynamic or static copula. It can serve to emphasise the relevance for risk management of appropriately modeling complex dependence structures. Finally we discuss the risk measures and how they affect the portfolio optimisation. We contend that more successful portfolio management can be achieved by combining extreme value analysis to describe downside tail risk and dynamic copulas to model nonlinear dependence structures. Conditional Value-at-Risk is adopted as pertinent measure of downside tail risk for portfolio optimisation. Using both realised portfolio returns and a set of out-of-sample Monte Carlo experiments, our novel portfolio strategy is confronted with the de facto mean-variance approach. The results suggest that the MV approach produces suboptimal portfolios or a less desirable risk-return tradeoff

Mathematical Analysis in Investment Theory: Applications to the Nigerian Stock Market

This thesis intends to optimise a portfolio of assets from the Nigerian Stock Exchange (NSE) using mathematical analysis in the investment theory to model the Nigerian financial market data better. In this work, we analysed the 82 stocks which were consistently traded in the NSE throughout 4years from August 2009 to August 2013. We attempt to maximise the expected return and minimise the variance of the portfolio by using Markowitz's portfolio selection model and a three-objective linear programming model allocating different percentages of weight to different assets to obtain an optimal/feasible portfolio of the financial sector of the NSM. The mean and the standard deviation served as constraints in the three-objective model used, and we constructed portfolios with the aims of maximising the returns and the Sharpe ratio and minimising the Standard Deviation (Variance) respectively. In another development, we use Random Matrix Theory (RMT) to analyse the Eigen-structure of the empirical correlations, apply the Marchenko-Pastur distribution of eigenvalues of a purely random matrix to investigate the presence of investment-pertinent information contained in the empirical correlation matrix of the selected stocks. We use a hypothesised standard normal distribution of eigenvector components from RMT to assess deviations of the empirical eigenvectors to this distribution for different eigenvalues. We also use the Inverse Participation Ratio to measure the deviation of eigenvectors of the empirical correlation matrix from RMT results. These preliminary results on the dynamics of asset price correlations in the NSE are essential for improving risk-return trade-offs associated with Markowitz's portfolio optimisation in the stock exchange, which we achieve by cleaning up the correlation matrix. Since the variance-covariance method underestimates risk, we employ Monte-Carlo simulations to estimate Value-at-Risk (VaR) and copula for a portfolio of 9 stocks of NSE. The result compared with historical simulation and variance-covariance data. Finally, with the outcome of our simulation and analysis, we were able to select the assets that form the optimal portfolio and the weights allocation to each stock. We were able to provide advice to the investors and market practitioners on how best to invest in the sector of NSE. We propose to measure the extent of closeness or otherwise in selected sectors of the NSE and the Johannesburg Stock Exchange (JSE) in our future work

Price Calibration of basket default swap: Evidence from Japanese market

The aim of this paper is the price calibration of basket default swap from Japanese market data. The value of this instruments depend on the number of factors including credit rating of the obligors in the basket, recovery rates, intensity of default, basket size and the correlation of obligors in the basket. A fundamental part of the pricing framework is the estimation of the instantaneous default probabilities for each obligor. Because default probabilities depend on the credit quality of the considered obligor, well-calibrated credit curves are a main ingredient for constructing default times. The calibration of credit curves take into account internal information on credit migrations and default history. We refer to Japan Credit Rating Agency to obtain rating transition matrix and cumulative default rates. Default risk is often considered as a rare-event and then, many studies have shown that many distributions have fatter tails than those captured by the normal distribution. Subsequently, the choice of copula and the choice of procedures for rare-event simulation govern the pricing of basket credit derivatives. Joshi and Kainth (2004) introduced an Importance Sampling technique for rare-event that forces a predetermined number of defaults to occur on each path. We consider using Gaussian copula and t-student copula and study their impact on basket credit derivative prices. We will present an application of the Canonical Maximum Likelihood Method (CML) for calibrating t-student copula to Japanese market data